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| {{Differential equations}}
| | I know that all of we have watched the term calorie several times over by this point, nevertheless how many of we have actually thought regarding what a calorie actually is? Probably not too many. When it absolutely comes down to it, calories are what aid you achieve or goals of either losing or gaining weight, so its important to recognize what a calorie really is and how it effects these goals.<br><br>On the other hand, overweight people who can [http://safedietplans.com/bmr-calculator bmr calculator] try to consume only 500 calories a day can almost likely be starving themselves. Because the body is more employed to taking in more than 2000 calories per day or even more, then the abrupt drop of calorie consumption usually signal the body into starvation mode. In this way, the body's metabolism will slow down to conserve energy.<br><br>Some would think which it is needless to employ the basal metabolic rate because a guide for controlling a daily calories. It's just as easy to limit oneself to 1200 to 1500 calories a day. This will be fine if we were all same. However, this one size fits all attitude of dieting might function for several, however, for the majority they can need something more that fits closer to their lifestyle.<br><br>Carbs, when converted to glucose, are used basically for power. Foods that are classified generally as carbs include: grains and their flours, potatoes, sugars (all forms), fruits, vegetables, and anything created from them.<br><br>Folks always ask how many calories could they be consuming in a day to keep there fat, youll really have to a small math. The initially step is calculating the bmr which this really is the amount of vitality the body takes in and must function properly. We use regarding 60% of the calories we consume everyday for the usual bodily functions like really by being alive plus breathing the others which influence your BMR are height, weight, age and sex.<br><br>Calories In is simple. This really is merely the amount of calories you eat plus drink every day, no matter where they come from. There are numerous ways to look these up. Fitday.com plus Calorie-Count.com are two good internet resources. You do need to track the Calories In. Fitday.com has tools for this, or you are able to create an Excel spreadsheet, or write them in a notebook. But you do it, keep track of the Calories In every day. As a side benefit, recognizing you'll have to write down which piece of cake assists motivate you to not eat it.<br><br>I hope you're in the usual range, but in the event you are overweight you are able to plan certain weight reduction system considering a BMR plus present activity level and hopefully enhance your health. |
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| In [[mathematics]], an '''autonomous system''' or '''autonomous differential equation''' is a [[simultaneous equations|system]] of [[ordinary differential equation]]s which does not explicitly depend on the [[independent variable]]. When the variable is time, they are also called [[time-invariant system]]s.
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| Many laws in [[physics]], where the independent variable is usually assumed to be [[time]], are expressed as autonomous systems because it is assumed the [[Physical law|laws of nature]] which hold now are identical to those for any point in the past or future.
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| Autonomous systems are closely related to [[dynamical system]]s. Any autonomous system can be transformed into a dynamical system and, using very weak assumptions, a dynamical system can be transformed into an autonomous system.
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| == Definition ==
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| An '''autonomous system''' is a [[system of ordinary differential equation]]s of the form
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| :<math>\frac{d}{dt}x(t)=f(x(t))</math>
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| where ''x'' takes values in ''n''-dimensional [[Euclidean space]] and ''t'' is usually time.
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| It is distinguished from systems of differential equations of the form
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| :<math>\frac{d}{dt}x(t)=g(x(t),t)</math>
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| in which the law governing the rate of motion of a particle depends not only on the particle's location, but also on time; such systems are not autonomous.
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| == Properties ==
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| Let <math>x_1(t)</math> be a unique solution of the
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| [[initial value problem]] for an autonomous system | |
| :<math>\frac{d}{dt}x(t)=f(x(t)) \, \mathrm{,} \quad x(0)=x_0</math>. | |
| Then <math>x_2(t)=x_1(t-t_0)</math> solves
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| :<math>\frac{d}{dt}x(t)=f(x(t)) \, \mathrm{,} \quad x(t_0)=x_0</math>.
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| Indeed, denoting <math>s=t-t_0</math> we have <math>x_1(s)=x_2(t)</math>
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| and <math>ds=dt</math>, thus
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| :<math>\frac{d}{dt}x_2(t)=\frac{d}{dt}x_1(t-t_0)=\frac{d}{ds}x_1(s)=
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| f(x_1(s))=f(x_2(t))</math>.
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| For the initial condition, the verification is trivial,
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| :<math>x_2(t_0)=x_1(t_0-t_0)=x_1(0)=x_0</math>.
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| ==Example==
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| The equation <math>y'=(2-y)y</math> is autonomous, since the independent variable,
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| let us call it <math>x</math>, does not explicitly appear in the equation.
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| To plot the [[slope field]] and [[isocline]] for this equation, one can use the following
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| code in [[GNU Octave]]/[[MATLAB]]
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| <source lang="matlab">
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| Ffun = @(X,Y)(2-Y).*Y; % function f(x,y)=(2-y)y
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| [X,Y]=meshgrid(0:.2:6,-1:.2:3); % choose the plot sizes
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| DY=Ffun(X,Y); DX=ones(size(DY)); % generate the plot values
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| quiver(X,Y,DX,DY); % plot the direction field
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| hold on;
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| contour(X,Y,DY,[0 2]); %add the isoclines
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| title('Slope field and isoclines for f(x,y)=(2-y)y')
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| </source>
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| One can observe from the plot that the function <math>(2-y)y</math> is <math>x</math>-invariant, and so is the shape of the solution, i.e. <math>y(x)=y(x-x_0)</math> for any shift <math>x_0</math>.
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| Solving the equation symbolically in [[MATLAB]], by running
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| <source lang="matlab">
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| y=dsolve('Dy=(2-y)*y','x'); % solve the equation symbolically
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| </source>
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| we obtain two [[Equilibrium point|equilibrium]] solutions, <math>y=0</math> and <math>y=2</math>,
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| and a third solution involving an unknown constant <math>C_3</math>, | |
| <source lang="matlab">
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| y(3)=-2/(exp(C3 - 2*x) - 1)
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| </source>
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| Picking up some specific values for the [[initial condition]], we can add the plot of several solutions
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| <source lang="matlab">
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| y1=dsolve('Dy=(2-y)*y','y(1)=1','x'); % solve the initial value problem symbolically
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| y2=dsolve('Dy=(2-y)*y','y(2)=1','x'); % for different initial conditions
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| y3=dsolve('Dy=(2-y)*y','y(3)=1','x'); y4=dsolve('Dy=(2-y)*y','y(1)=3','x');
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| y5=dsolve('Dy=(2-y)*y','y(2)=3','x'); y6=dsolve('Dy=(2-y)*y','y(3)=3','x');
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| ezplot(y1, [0 6]); ezplot(y2, [0 6]); % plot the solutions
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| ezplot(y3, [0 6]); ezplot(y4, [0 6]); ezplot(y5, [0 6]); ezplot(y6, [0 6]);
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| title('Slope field, isoclines and solutions for f(x,y)=(2-y)y')
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| </source>
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| == Qualitative analysis ==
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| Autonomous systems can be analyzed qualitatively using the [[phase space]]; in the one-variable case, this is the [[phase line (mathematics)|phase line]].
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| == Solution techniques ==
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| The following techniques apply to one-dimensional autonomous differential equations. Any one-dimensional equation of order <math>n</math> is equivalent to an <math>n</math>-dimensional first-order system (as described in [[Ordinary differential equation#Reduction to a first order system]]), but not necessarily vice versa.
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| === First order ===
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| The first-order autonomous equation | |
| :<math>\frac{dx}{dt} = f(x)</math>
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| is [[Examples of differential equations#Separable first order linear ordinary differential equations|separable]], so it can easily be solved by rearranging it into the integral form | |
| :<math>t + C = \int \frac{dx}{f(x)}</math>
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| === Second order ===
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| The second-order autonomous equation
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| :<math>\frac{d^2x}{dt^2} = f(x, x')</math>
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| is more difficult, but it can be solved<ref>{{cite book |last=Boyce |first=William E. |coauthors=Richard C. DiPrima |title=Elementary Differential Equations and Boundary Volume Problems |edition=8th ed. |year=2005 |publisher=John Wiley & Sons |isbn=0-471-43338-1 |pages=133}}</ref> by introducing the new variable
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| :<math>v = \frac{dx}{dt}</math>
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| and expressing the [[second derivative]] of <math>x</math> (via the [[chain rule]]) as
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| :<math>\frac{d^2x}{dt^2} = \frac{dv}{dt} = \frac{dx}{dt}\frac{dv}{dx} = v\frac{dv}{dx}</math>
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| so that the original equation becomes
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| :<math>v\frac{dv}{dx} = f(x, v)</math>
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| which is a first order equation containing no reference to the independent variable <math>t</math> and if solved provides <math>v</math> as a function of <math>x</math>. Then, recalling the definition of <math>v</math>:
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| :<math>\frac{dx}{dt} = v(x) \quad \Rightarrow \quad t + C = \int \frac{d x}{v(x)} </math>
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| which is an implicit solution.
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| ====Special case: ''x''<nowiki>''</nowiki> = ''f''(''x'')====
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| The special case where <math>f</math> is independent of <math>x'</math>
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| :<math>\frac{d^2 x}{d t^2} = f(x)</math>
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| benefits from separate treatment.<ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0301.pdf Second order autonomous equation] at [[eqworld]].</ref> These types of equations are very common in [[classical mechanics]] because they are always [[Hamiltonian system]]s.
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| The idea is to make use of the identity (barring [[division by zero]] issues)
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| :<math>\frac{d x}{d t} = \left(\frac{d t}{d x}\right)^{-1}</math>
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| which follows from the [[chain rule]]. Note aside then that by inverting both sides of a first order autonomous system, one can immediately integrate with respect to <math>x</math>:
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| :<math>\frac{d x}{d t} = f(x) \quad \Rightarrow \quad \frac{d t}{d x} = \frac{1}{f(x)} \quad \Rightarrow \quad t + C = \int \frac{dx}{f(x)}</math>
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| which is another way to view the separation of variables technique. A natural question is then: can we do something like this with higher order equations? The answer is yes for second order equations, but there's more work to do. The second derivative must be expressed as a derivative with respect to <math>x</math> instead of <math>t</math>:
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| :<math>
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| \begin{align}
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| \frac{d^2 x}{d t^2} &= \frac{d}{d t}\left(\frac{d x}{d t}\right) =
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| \frac{d}{d x}\left(\frac{d x}{d t}\right) \frac{d x}{d t} =
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| \frac{d}{d x}\left(\left(\frac{d t}{d x}\right)^{-1}\right) \left(\frac{d t}{d x}\right)^{-1} = \\
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| & = - \left(\frac{d t}{d x}\right)^{-2} \frac{d^2 t}{d x^2} \left(\frac{d t}{d x}\right)^{-1} =
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| - \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \\
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| & = \frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right)
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| \end{align}
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| </math>
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| To reemphasize: what's been accomplished is that the second derivative in <math>t</math> has been expressed as a derivative in <math>x</math>. The original second order equation may then finally be integrated:
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| :<math>\frac{d^2 x}{d t^2} = f(x)</math>
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| :<math>\frac{d}{d x}\left(\frac{1}{2}\left(\frac{d t}{d x}\right)^{-2}\right) = f(x)</math>
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| :<math>\left(\frac{d t}{d x}\right)^{-2} = 2 \int f(x) dx + C_1</math>
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| :<math>\frac{d t}{d x} = \pm \frac{1}{\sqrt{2 \int f(x) dx + C_1}}</math>
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| :<math>t + C_2 = \pm \int \frac{dx}{\sqrt{2 \int f(x) dx + C_1}}</math>
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| This is an implicit solution, and beyond that the greatest potential problem is inability to simplify the integrals, which implies difficulty or impossibility in evaluating the integration constants.
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| ====Special case: ''x''<nowiki>''</nowiki> = ''x''<nowiki>'</nowiki><sup>n</sup> ''f''(''x'')====
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| Using the above mentality, we can extend the technique to the more general equation
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| :<math>\frac{d^2 x}{d t^2} = \left(\frac{d x}{d t}\right)^n f(x)</math>
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| where <math>n</math> is some parameter not equal to two. This will work since the second derivative can be written in a form involving a power of <math>x'</math>. Rewriting the second derivative, rearranging, and expressing the left side as a derivative:
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| :<math>- \left(\frac{d t}{d x}\right)^{-3} \frac{d^2 t}{d x^2} = \left(\frac{d t}{d x}\right)^{-n} f(x)</math>
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| :<math>- \left(\frac{d t}{d x}\right)^{n - 3} \frac{d^2 t}{d x^2} = f(x)</math>
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| :<math>\frac{d}{d x}\left(\frac{1}{2 - n}\left(\frac{d t}{d x}\right)^{n - 2}\right) = f(x)</math>
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| :<math>\left(\frac{d t}{d x}\right)^{n - 2} = (2 - n) \int f(x) dx + C_1</math>
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| :<math>t + C_2 = \int \left((2 - n) \int f(x) dx + C_1\right)^{\frac{1}{n - 2}} dx</math>
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| The right will carry +/- if <math>n</math> is even. The treatment must be different if <math>n = 2</math>:
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| :<math>- \left(\frac{d t}{d x}\right)^{-1} \frac{d^2 t}{d x^2} = f(x)</math>
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| :<math>-\frac{d}{d x}\left(\ln\left(\frac{d t}{d x}\right)\right) = f(x)</math>
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| :<math>\frac{d t}{d x} = C_1 e^{-\int f(x) dx}</math>
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| :<math>t + C_2 = C_1 \int e^{-\int f(x) dx} dx</math>
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| === Higher orders ===
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| There is no analogous method for solving third- or higher-order autonomous equations. Such equations can only be solved exactly if they happen to have some other simplifying property, for instance [[linear differential equation|linearity]] or dependence of the right side of the equation on the dependent variable only<ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0503.pdf Third order autonomous equation] at [[eqworld]].</ref><ref>[http://eqworld.ipmnet.ru/en/solutions/ode/ode0506.pdf Fourth order autonomous equation] at [[eqworld]].</ref> (i.e., not its derivatives). This should not be surprising, considering that nonlinear autonomous systems in three dimensions can produce truly [[chaos theory|chaotic]] behavior such as the [[Lorenz attractor]] and the [[Rössler attractor]].
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| With this mentality, it also isn't too surprising that general non-autonomous equations of second order can't be solved explicitly, since these can also be chaotic (an example of this is a periodically forced pendulum<ref>{{cite book
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| | last = Blanchard, Devaney, Hall
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| | title = Differential Equations
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| | publisher = Brooks/Cole Publishing Co
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| | year = 2005
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| | pages = 540–543
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| | isbn = 0-495-01265-3}}</ref>).
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| == See also ==
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| * [[Time-invariant system]]
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| * [[Non-autonomous system (mathematics)]]
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| == References ==
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| <references />
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| {{DEFAULTSORT:Autonomous System (Mathematics)}}
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| [[Category:Differential equations]]
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| [[Category:Dynamical systems]]
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| [[Category:Ordinary differential equations]]
| |
I know that all of we have watched the term calorie several times over by this point, nevertheless how many of we have actually thought regarding what a calorie actually is? Probably not too many. When it absolutely comes down to it, calories are what aid you achieve or goals of either losing or gaining weight, so its important to recognize what a calorie really is and how it effects these goals.
On the other hand, overweight people who can bmr calculator try to consume only 500 calories a day can almost likely be starving themselves. Because the body is more employed to taking in more than 2000 calories per day or even more, then the abrupt drop of calorie consumption usually signal the body into starvation mode. In this way, the body's metabolism will slow down to conserve energy.
Some would think which it is needless to employ the basal metabolic rate because a guide for controlling a daily calories. It's just as easy to limit oneself to 1200 to 1500 calories a day. This will be fine if we were all same. However, this one size fits all attitude of dieting might function for several, however, for the majority they can need something more that fits closer to their lifestyle.
Carbs, when converted to glucose, are used basically for power. Foods that are classified generally as carbs include: grains and their flours, potatoes, sugars (all forms), fruits, vegetables, and anything created from them.
Folks always ask how many calories could they be consuming in a day to keep there fat, youll really have to a small math. The initially step is calculating the bmr which this really is the amount of vitality the body takes in and must function properly. We use regarding 60% of the calories we consume everyday for the usual bodily functions like really by being alive plus breathing the others which influence your BMR are height, weight, age and sex.
Calories In is simple. This really is merely the amount of calories you eat plus drink every day, no matter where they come from. There are numerous ways to look these up. Fitday.com plus Calorie-Count.com are two good internet resources. You do need to track the Calories In. Fitday.com has tools for this, or you are able to create an Excel spreadsheet, or write them in a notebook. But you do it, keep track of the Calories In every day. As a side benefit, recognizing you'll have to write down which piece of cake assists motivate you to not eat it.
I hope you're in the usual range, but in the event you are overweight you are able to plan certain weight reduction system considering a BMR plus present activity level and hopefully enhance your health.